On the 2D plane strain problem for a harmonic stress applied to an impervious elastic layer resting on a porous elastic half space

Abstract

We solve here for stresses and pore pressure induced in a medium comprising of an impervious elastic layer resting on a water saturated porous elastic half space when the upper surface of the layer is acted upon by a normal stress field varying harmonically in time. The stress is constant along one horizontal space direction while it varies in a prescribed manner along the perpendicular horizontal direction. A formal solution for this boundary value problem is obtained using the Fourier transform approach. We consider for illustration a case where the layer and the half space have nominal values for elastic constants, hydraulic diffusivity and Skempton's coefficient. The stress field imposed on the layer surface acts over a finite width, comparable to layer thickness, and has a temporal variation with a period of one year. The resultant induced stresses in the layer and the induced stresses and pore pressure in the half space have the same temporal period as that of the applied stress. Their amplitudes and magnitudes of the shifts in their respective phases relative to that of the surface stress decrease with distance away from the immediate vicinity of the applied stress. The phase shifts in the induced pore pressure represent the effect of volume changes of solid material, pore spaces and water in the half space under the induced stresses as well as diffusion of water under these changes. The phase shifts in induced stresses arise because they are influenced in turn by changes in induced pore pressure as prescribed in Biot's and Rice and Cleary's theory for porous elastic media. The computed magnitudes of the phase shifts are small because of the chosen values of the parameters for the layer and the half space as well as the relatively long period of variation of the applied stress. The analysis has possible application in the field of reservoir induced seismicity in at least a few cases where, on local site investigations, such a medium geometry would appear relevant in the first approximation.